comm_alg/CommAlg/krull.lean

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import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/- This file contains the definitions of height of an ideal, and the krull
dimension of a commutative ring.
There are also sorried statements of many of the theorems that would be
really nice to prove.
I'm imagining for this file to ultimately contain basic API for height and
krull dimension, and the theorems will probably end up other files,
depending on how long the proofs are, and what extra API needs to be
developed.
-/
/-- If something is smaller that Bot of a PartialOrder after attaching another Bot, it must be Bot. -/
lemma lt_bot_eq_WithBot_bot [PartialOrder α] [OrderBot α] {a : WithBot α} (h : a < (⊥ : α)) : a = ⊥ := by
cases a
. rfl
. cases h.not_le (WithBot.coe_le_coe.2 bot_le)
namespace Ideal
open LocalRing
variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
/-- Definitions -/
noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
noncomputable def krullDim (R : Type _) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
noncomputable def codimension (J : Ideal R) : WithBot ℕ∞ := ⨅ I ∈ {I : PrimeSpectrum R | J ≤ I.asIdeal}, height I
lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
lemma krullDim_def (R : Type _) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
lemma krullDim_def' (R : Type _) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
/-- A lattice structure on WithBot ℕ∞. -/
noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
/-- Height of ideals is monotonic. -/
lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := by
apply Set.chainHeight_mono
intro J' hJ'
show J' < J
exact lt_of_lt_of_le hJ' I_le_J
@[simp]
lemma krullDim_le_iff {R : Type _} [CommRing R] {n : } :
krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
@[simp]
lemma krullDim_le_iff' {R : Type _} [CommRing R] {n : ℕ∞} :
krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
@[simp]
lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I
/-- In a domain, the height of a prime ideal is Bot (0 in this case) iff it's the Bot ideal. -/
@[simp]
lemma height_zero_iff_bot {D: Type _} [CommRing D] [IsDomain D] {P : PrimeSpectrum D} : height P = 0 ↔ P = ⊥ := by
constructor
· intro h
unfold height at h
simp only [Set.chainHeight_eq_zero_iff] at h
apply eq_bot_of_minimal
intro I
by_contra x
have : I ∈ {J | J < P} := x
rw [h] at this
contradiction
· intro h
unfold height
simp only [bot_eq_zero', Set.chainHeight_eq_zero_iff]
by_contra spec
change _ ≠ _ at spec
rw [← Set.nonempty_iff_ne_empty] at spec
obtain ⟨J, JlP : J < P⟩ := spec
have JneP : J ≠ P := ne_of_lt JlP
rw [h, ←bot_lt_iff_ne_bot, ←h] at JneP
have := not_lt_of_lt JneP
contradiction
/-- The Krull dimension of a ring being ≥ n is equivalent to there being an
ideal of height ≥ n. -/
@[simp]
lemma le_krullDim_iff {R : Type _} [CommRing R] {n : } :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by
constructor
· unfold krullDim
intro H
by_contra H1
push_neg at H1
by_cases n ≤ 0
· rw [Nat.le_zero] at h
rw [h] at H H1
have : ∀ (I : PrimeSpectrum R), ↑(height I) = (⊥ : WithBot ℕ∞) := by
intro I
specialize H1 I
exact lt_bot_eq_WithBot_bot H1
rw [←iSup_eq_bot] at this
have := le_of_le_of_eq H this
rw [le_bot_iff] at this
exact WithBot.coe_ne_bot this
· push_neg at h
have : (n: ℕ∞) > 0 := Nat.cast_pos.mpr h
replace H1 : ∀ (I : PrimeSpectrum R), height I ≤ n - 1 := by
intro I
specialize H1 I
apply ENat.le_of_lt_add_one
rw [←ENat.coe_one, ←ENat.coe_sub, ←ENat.coe_add, tsub_add_cancel_of_le]
exact WithBot.coe_lt_coe.mp H1
exact h
replace H1 : ∀ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) ≤ ↑(n - 1):=
fun _ ↦ (WithBot.coe_le rfl).mpr (H1 _)
rw [←iSup_le_iff] at H1
have : ((n : ℕ∞) : WithBot ℕ∞) ≤ (((n - 1 : ) : ℕ∞) : WithBot ℕ∞) := le_trans H H1
norm_cast at this
have that : n - 1 < n := by refine Nat.sub_lt h (by norm_num)
apply lt_irrefl (n-1) (trans that this)
· rintro ⟨I, h⟩
have : height I ≤ krullDim R := by apply height_le_krullDim
exact le_trans h this
#check ENat.recTopCoe
/- terrible place for these two lemmas. Also this probably exists somewhere
Also this is a terrible proof
-/
lemma eq_top_iff' (n : ℕ∞) : n = ↔ ∀ m : , m ≤ n := by
refine' ⟨fun a b => _, fun h => _⟩
. rw [a]; exact le_top
. induction' n using ENat.recTopCoe with n
. rfl
. exfalso
apply not_lt_of_ge (h (n + 1))
norm_cast
norm_num
lemma eq_top_iff (n : WithBot ℕ∞) : n = ↔ ∀ m : , m ≤ n := by
aesop
induction' n using WithBot.recBotCoe with n
. exfalso
have := (a 0)
simp [not_lt_of_ge] at this
induction' n using ENat.recTopCoe with n
. rfl
. have := a (n + 1)
exfalso
change (((n + 1) : ℕ∞) : WithBot ℕ∞) ≤ _ at this
simp [WithBot.coe_le_coe] at this
change ((n + 1) : ℕ∞) ≤ (n : ℕ∞) at this
simp [ENat.add_one_le_iff] at this
lemma krullDim_eq_top_iff (R : Type _) [CommRing R] :
krullDim R = ↔ ∀ (n : ), ∃ I : PrimeSpectrum R, n ≤ height I := by
simp_rw [eq_top_iff, le_krullDim_iff]
change (∀ (m : ), ∃ I, ((m : ℕ∞) : WithBot ℕ∞) ≤ height I) ↔ _
simp [WithBot.coe_le_coe]
/-- The Krull dimension of a local ring is the height of its maximal ideal. -/
lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) := by
apply le_antisymm
. rw [krullDim_le_iff']
intro I
exact WithBot.coe_mono <| height_le_of_le <| le_maximalIdeal I.2.1
. simp only [height_le_krullDim]
/-- The height of a prime `𝔭` is greater than `n` if and only if there is a chain of primes less than `𝔭`
with length `n + 1`. -/
lemma lt_height_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
n < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
induction' n using ENat.recTopCoe with n
. simp
. rw [←(ENat.add_one_le_iff <| ENat.coe_ne_top _)]
show ((↑(n + 1):ℕ∞) ≤ _) ↔ ∃c, _ ∧ _ ∧ ((_ : WithTop ) = (_:ℕ∞))
rw [Ideal.height, Set.le_chainHeight_iff]
show (∃ c, (List.Chain' _ c ∧ ∀𝔮, 𝔮 ∈ c → 𝔮 < 𝔭) ∧ _) ↔ _
norm_cast
simp_rw [and_assoc]
/-- Form of `lt_height_iff''` for rewriting with the height coerced to `WithBot ℕ∞`. -/
lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
(n : WithBot ℕ∞) < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
rw [WithBot.coe_lt_coe]
exact lt_height_iff'
#check height_le_krullDim
--some propositions that would be nice to be able to eventually
/-- The prime spectrum of the zero ring is empty. -/
lemma primeSpectrum_empty_of_subsingleton [Subsingleton R] : IsEmpty <| PrimeSpectrum R where
false x := x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem
/-- A CommRing has empty prime spectrum if and only if it is the zero ring. -/
lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
constructor <;> contrapose
. rw [not_isEmpty_iff, ←not_nontrivial_iff_subsingleton, not_not]
apply PrimeSpectrum.instNonemptyPrimeSpectrum
. intro hneg h
exact hneg primeSpectrum_empty_of_subsingleton
/-- A ring has Krull dimension -∞ if and only if it is the zero ring -/
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
rw [Ideal.krullDim, ←primeSpectrum_empty_iff, iSup_eq_bot]
constructor <;> intro h
. rw [←not_nonempty_iff]
rintro ⟨a, ha⟩
cases h ⟨a, ha⟩
. rw [h.forall_iff]
trivial
/-- Nonzero rings have Krull dimension in ℕ∞ -/
lemma krullDim_nonneg_of_nontrivial (R : Type _) [CommRing R] [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by
have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
lift (Ideal.krullDim R) to ℕ∞ using h with k
use k
/-- An ideal which is less than a prime is not a maximal ideal. -/
lemma not_maximal_of_lt_prime {p : Ideal R} {q : Ideal R} (hq : IsPrime q) (h : p < q) : ¬IsMaximal p := by
intro hp
apply IsPrime.ne_top hq
apply (IsCoatom.lt_iff hp.out).mp
exact h
/-- Krull dimension is ≤ 0 if and only if all primes are maximal. -/
lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
show ((_ : WithBot ℕ∞) ≤ (0 : )) ↔ _
rw [krullDim_le_iff]
constructor <;> intro h I
. contrapose! h
have ⟨𝔪, h𝔪⟩ := I.asIdeal.exists_le_maximal (IsPrime.ne_top I.IsPrime)
let 𝔪p := (⟨𝔪, IsMaximal.isPrime h𝔪.1⟩ : PrimeSpectrum R)
have hstrct : I < 𝔪p := by
apply lt_of_le_of_ne h𝔪.2
intro hcontr
rw [hcontr] at h
exact h h𝔪.1
use 𝔪p
show (0 : ℕ∞) < (_ : WithBot ℕ∞)
rw [lt_height_iff'']
use [I]
constructor
. exact List.chain'_singleton _
. constructor
. intro I' hI'
simp only [List.mem_singleton] at hI'
rwa [hI']
. simp only [List.length_singleton, Nat.cast_one, zero_add]
. contrapose! h
change (0 : ℕ∞) < (_ : WithBot ℕ∞) at h
rw [lt_height_iff''] at h
obtain ⟨c, _, hc2, hc3⟩ := h
norm_cast at hc3
rw [List.length_eq_one] at hc3
obtain ⟨𝔮, h𝔮⟩ := hc3
use 𝔮
specialize hc2 𝔮 (h𝔮 ▸ (List.mem_singleton.mpr rfl))
apply not_maximal_of_lt_prime I.IsPrime
exact hc2
/-- For a nonzero ring, Krull dimension is 0 if and only if all primes are maximal. -/
lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
rw [←dim_le_zero_iff]
obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
have : n ≥ 0 := zero_le n
change _ ≤ _ at this
rw [←WithBot.coe_le_coe,←hn] at this
change (0 : WithBot ℕ∞) ≤ _ at this
constructor <;> intro h'
. rw [h']
. exact le_antisymm h' this
/-- In a field, the unique prime ideal is the zero ideal. -/
@[simp]
lemma field_prime_bot {K: Type _} [Field K] {P : Ideal K} : IsPrime P ↔ P = ⊥ := by
refine' ⟨fun primeP => Or.elim (eq_bot_or_top P) _ _, fun botP => _⟩
· intro P_top; exact P_top
. intro P_bot; exact False.elim (primeP.ne_top P_bot)
· rw [botP]
exact bot_prime
/-- In a field, all primes have height 0. -/
lemma field_prime_height_zero {K: Type _} [Nontrivial K] [Field K] (P : PrimeSpectrum K) : height P = 0 := by
have : IsPrime P.asIdeal := P.IsPrime
rw [field_prime_bot] at this
have : P = ⊥ := PrimeSpectrum.ext P ⊥ this
rwa [height_zero_iff_bot]
/-- The Krull dimension of a field is 0. -/
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim
simp only [field_prime_height_zero, ciSup_unique]
/-- A domain with Krull dimension 0 is a field. -/
lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
by_contra x
rw [Ring.not_isField_iff_exists_prime] at x
obtain ⟨P, ⟨h1, primeP⟩⟩ := x
let P' : PrimeSpectrum D := PrimeSpectrum.mk P primeP
have h2 : P' ≠ ⊥ := by
by_contra a
have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
contradiction
have pos_height : ¬ (height P') ≤ 0 := by
have : ⊥ ∈ {J | J < P'} := Ne.bot_lt h2
have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
unfold height
rw [←Set.one_le_chainHeight_iff] at this
exact not_le_of_gt (ENat.one_le_iff_pos.mp this)
have nonpos_height : height P' ≤ 0 := by
have := height_le_krullDim P'
rw [h] at this
aesop
contradiction
/-- A domain has Krull dimension 0 if and only if it is a field. -/
lemma domain_dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
constructor
· exact domain_dim_zero.isField
· intro fieldD
let h : Field D := fieldD.toField
exact dim_field_eq_zero
#check Ring.DimensionLEOne
-- This lemma is false!
lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry
/-- The converse of this is false, because the definition of "dimension ≤ 1" in mathlib
applies only to dimension zero rings and domains of dimension 1. -/
lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ 1 := by
show _ → ((_ : WithBot ℕ∞) ≤ (1 : ))
rw [krullDim_le_iff]
intro H p
apply le_of_not_gt
intro h
rcases (lt_height_iff''.mp h) with ⟨c, ⟨hc1, hc2, hc3⟩⟩
norm_cast at hc3
rw [List.chain'_iff_get] at hc1
specialize hc1 0 (by rw [hc3]; simp only)
set q0 : PrimeSpectrum R := List.get c { val := 0, isLt := _ }
set q1 : PrimeSpectrum R := List.get c { val := 1, isLt := _ }
specialize hc2 q1 (List.get_mem _ _ _)
change q0.asIdeal < q1.asIdeal at hc1
have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
exact (not_maximal_of_lt_prime p.IsPrime hc2) H
/-- The Krull dimension of a PID is at most 1. -/
lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by
rw [dim_le_one_iff]
exact Ring.DimensionLEOne.principal_ideal_ring R
private lemma singleton_chainHeight_le_one {α : Type _} {x : α} [Preorder α] : Set.chainHeight {x} ≤ 1 := by
unfold Set.chainHeight
simp only [iSup_le_iff, Nat.cast_le_one]
intro L h
unfold Set.subchain at h
simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at h
rcases L with (_ | ⟨a,L⟩)
. simp only [List.length_nil, zero_le]
rcases L with (_ | ⟨b,L⟩)
. simp only [List.length_singleton, le_refl]
simp only [List.chain'_cons, List.find?, List.mem_cons, forall_eq_or_imp] at h
rcases h with ⟨⟨h1, _⟩, ⟨rfl, rfl, _⟩⟩
exact absurd h1 (lt_irrefl _)
/-- The ring of polynomials over a field has dimension one. -/
lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
rw [le_antisymm_iff]
let X := @Polynomial.X K _
constructor
· unfold krullDim
apply @iSup_le (WithBot ℕ∞) _ _ _ _
intro I
have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
by_cases I = ⊥
· rw [← height_zero_iff_bot] at h
simp only [WithBot.coe_le_one, ge_iff_le]
rw [h]
exact bot_le
· push_neg at h
have : I.asIdeal ≠ ⊥ := by
by_contra a
have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
contradiction
have maxI := IsPrime.to_maximal_ideal this
have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
intro P
constructor
· intro H
simp only [Set.mem_setOf_eq] at H
by_contra x
push_neg at x
have : P.asIdeal ≠ ⊥ := by
by_contra a
have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
contradiction
have maxP := IsPrime.to_maximal_ideal this
have IneTop := IsMaximal.ne_top maxI
have : P ≤ I := le_of_lt H
rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
have : P = I := PrimeSpectrum.ext P I this
replace H : P ≠ I := ne_of_lt H
contradiction
· intro pBot
simp only [Set.mem_setOf_eq, pBot]
exact lt_of_le_of_ne bot_le h.symm
replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
unfold height
rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_chainHeight_le_one
· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
· obtain ⟨I, h⟩ := this
have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I
exact le_trans h this
have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
have : IsPrime (span {X}) := by
refine (span_singleton_prime ?hp).mpr primeX
exact Polynomial.X_ne_zero
let P := PrimeSpectrum.mk (span {X}) this
unfold height
use P
have : ⊥ ∈ {J | J < P} := by
simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot]
suffices : P.asIdeal ≠ ⊥
· by_contra x
rw [PrimeSpectrum.ext_iff] at x
contradiction
by_contra x
simp only [span_singleton_eq_bot] at x
have := @Polynomial.X_ne_zero K _ _
contradiction
have : {J | J < P}.Nonempty := Set.nonempty_of_mem this
rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this
-- lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
-- krullDim R + 1 = krullDim (Polynomial R) := sorry
lemma krull_height_theorem [Nontrivial R] [IsNoetherianRing R] (P: PrimeSpectrum R) (S: Finset R)
(h: P.asIdeal ∈ Ideal.minimalPrimes (Ideal.span S)) : height P ≤ S.card := by
sorry
lemma dim_mvPolynomial [Field K] (n : ) : krullDim (MvPolynomial (Fin n) K) = n := sorry
lemma height_eq_dim_localization :
height I = krullDim (Localization.AtPrime I.asIdeal) := sorry
lemma height_add_dim_quotient_le_dim : height I + krullDim (R I.asIdeal) ≤ krullDim R := sorry